Unscented statistical linearization and robustified Kalman filter for nonlinear systems with parameter uncertainties

One of the most significant systems that can be expressed by partial differential equations (PDEs) is the transmission pipeline system. To avoid the accidents that originated from oil and gas pipeline leakage, the exact location and quantity of leakage are required to be recognized. The designed goal is a leakage diagnosis based on the system model and the use of real data provided by transmission line systems. Nonlinear equations of the system have been extracted employing continuity and momentum equations. In this paper, the extended Kalman filter (EKF) is used to detect and locate the leakage and to attenuate the negative effects of measurement and process noises. Besides, a robust extended Kalman filter (REKF) is applied to compensate for the effect of parameter uncertainty. The quantity and the location of the occurred leakage are estimated along the pipeline. Simulation results show that REKF has better estimations of the leak and its location as compared with that of EKF. This filter is robust against process noise, measurement noise, parameter uncertainties, and guarantees a higher limit for the covariance of state estimation error as well. It is remarkable that simulation results are evaluated by OLGA software.

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Robust Nonlinear Kalman Filters for Nonlinear Systems with Parameter Uncertainties

Transactions of the Society of Instrument and Control Engineers ◽

10.9746/sicetr.53.527 ◽

2017 ◽

Vol 53 (10) ◽

pp. 527-538

Author(s):

Shinji ISHIHARA ◽

Masaki YAMAKITA

Keyword(s):

Nonlinear Systems ◽

Kalman Filters ◽

Parameter Uncertainties

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Tracking: B

Probability in Electrical Engineering and Computer Science ◽

10.1007/978-3-030-49995-2_10 ◽

2021 ◽

pp. 193-204

Author(s):

Jean Walrand

Keyword(s):

Kalman Filter ◽

Nonlinear Systems ◽

Extended Kalman Filter ◽

Random Variable

AbstractIn Chapter Tracking: A, we explained the estimation of a random variable based on observations. We also described the Kalman filter and we gave a number of examples. In this chapter, we derive the Kalman filter and explain some of its properties. We also discuss the extended Kalman filter.Section 10.1 explains how to update an estimate as one makes additional observations. Section 10.2 derives the Kalman filter. The properties of the Kalman filter are explained in Sect. 10.3. Section 10.4 shows how the Kalman filter is extended to nonlinear systems.

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Adaptive Unscented Kalman Filter for Target Tracking in the Presence of Nonlinear Systems Involving Model Mismatches

Remote Sensing ◽

10.3390/rs9070657 ◽

2017 ◽

Vol 9 (7) ◽

pp. 657 ◽

Cited By ~ 8

Author(s):

Huan Zhou ◽

Hanqiao Huang ◽

Hui Zhao ◽

Xin Zhao ◽

Xiang Yin

Keyword(s):

Kalman Filter ◽

Nonlinear Systems ◽

Target Tracking ◽

Unscented Kalman Filter ◽

Adaptive Unscented Kalman Filter

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An EKF-Based Fixed-Point Iterative Filter for Nonlinear Systems

Sensors ◽

10.3390/s19081893 ◽

2019 ◽

Vol 19 (8) ◽

pp. 1893

Author(s):

Feng ◽

Wen

Keyword(s):

Fixed Point ◽

Kalman Filter ◽

Nonlinear Systems ◽

Iterative Method ◽

Taylor Series ◽

Nonlinear Filtering ◽

Higher Order ◽

Point Function ◽

State Estimate ◽

Linearized System

In this paper, a fixed-point iterative filter developed from the classical extended Kalman filter (EKF) was proposed for general nonlinear systems. As a nonlinear filter developed from EKF, the state estimate was obtained by applying the Kalman filter to the linearized system by discarding the higher-order Taylor series items of the original nonlinear system. In order to reduce the influence of the discarded higher-order Taylor series items and improve the filtering accuracy of the obtained state estimate of the steady-state EKF, a fixed-point function was solved though a nested iterative method, which resulted in a fixed-point iterative filter. The convergence of the fixed-point function is also discussed, which provided the existing conditions of the fixed-point iterative filter. Then, Steffensen’s iterative method is presented to accelerate the solution of the fixed-point function. The final simulation is provided to illustrate the feasibility and the effectiveness of the proposed nonlinear filtering method.

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Optimal control of nonlinear systems using RBF neural network and adaptive extended Kalman filter

2009 American Control Conference ◽

10.1109/acc.2009.5160105 ◽

2009 ◽

Cited By ~ 14

Author(s):

Peda V. Medagam ◽

Farzad Pourboghrat

Keyword(s):

Neural Network ◽

Optimal Control ◽

Kalman Filter ◽

Nonlinear Systems ◽

Extended Kalman Filter ◽

Rbf Neural Network ◽

Adaptive Extended Kalman Filter

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M-estimator based robust unscented Kalman filter through statistical linearization

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331218793925 ◽

2018 ◽

Vol 41 (7) ◽

pp. 2016-2025 ◽

Cited By ~ 1

Author(s):

Guobin Chang ◽

Tianhe Xu ◽

Haitao Wang

Keyword(s):

Kalman Filter ◽

Gaussian Distribution ◽

Unscented Kalman Filter ◽

Weighted Least Squares ◽

Point Of View ◽

Tuning Parameter ◽

Statistical Linearization ◽

Unscented Transformation ◽

Actual Distribution ◽

M Estimator

The robust unscented Kalman filter (UKF) is revisited in this paper from a new point of view, namely the statistical linear regression (SLR) perspective of the unscented transformation (UT). When the actual distribution of the observation noise deviates from the assumed Gaussian distribution, the performance of the filter may degrade significantly owing to lacking robustness. After linearizing the nonlinear observation equation using the SLR perspective, the M-estimator is employed to replace the weighted least-squares one, resulting in the robust UKF. Besides providing new theoretical aspects, the advantageous performance of the proposed filter is re-emphasized in terms of the following three: (1) robust, the proposed filter is robust against deviations from the assumed Gaussian distribution; (2) efficient, by elaborately selecting the tuning parameter of the M-estimator, the robust filter will have a slight efficiency loss compared with the UKF when the Gaussian assumption does hold; (3) accurate, the proposed filter achieves robustness without compromising the accuracy of the UT.

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Ellipsoidal and Gaussian Kalman Filter Model for Discrete-Time Nonlinear Systems

Mathematics ◽

10.3390/math7121168 ◽

2019 ◽

Vol 7 (12) ◽

pp. 1168 ◽

Cited By ~ 2

Author(s):

Ligang Sun ◽

Hamza Alkhatib ◽

Boris Kargoll ◽

Vladik Kreinovich ◽

Ingo Neumann

Keyword(s):

Kalman Filter ◽

Nonlinear Systems ◽

State Estimation ◽

Discrete Time ◽

Probability Distributions ◽

Nonlinear Problems ◽

The State ◽

Filter Model ◽

Highly Nonlinear ◽

Correction Step

In this paper, we propose a new technique—called Ellipsoidal and Gaussian Kalman filter—for state estimation of discrete-time nonlinear systems in situations when for some parts of uncertainty, we know the probability distributions, while for other parts of uncertainty, we only know the bounds (but we do not know the corresponding probabilities). Similarly to the usual Kalman filter, our algorithm is iterative: on each iteration, we first predict the state at the next moment of time, and then we use measurement results to correct the corresponding estimates. On each correction step, we solve a convex optimization problem to find the optimal estimate for the system’s state (and the optimal ellipsoid for describing the systems’s uncertainty). Testing our algorithm on several highly nonlinear problems has shown that the new algorithm performs the extended Kalman filter technique better—the state estimation technique usually applied to such nonlinear problems.

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